Atlas › Modular forms › Weight 4 › Level 221 › 221.4.a › 221.4.a.b

Newform orbit 221.4.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 3q^{2} - 4q^{3} + q^{4} - 12q^{5} - 40q^{7} + 18q^{8} + 24q^{9} - 75q^{10} - 34q^{11} + 35q^{12} + 130q^{13} - 100q^{14} - 230q^{15} - 139q^{16} - 170q^{17} - 152q^{18} - 436q^{19} - 247q^{20} - 186q^{21} - 119q^{22} + 26q^{23} - 87q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} + 7x^{9} - 163x^{8} - 808x^{7} + 8846x^{6} + 23378x^{5} - 197146x^{4} - 104568x^{3} + 1344417x^{2} - 805049x - 914611$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
17	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−3	−4	−12	−40	−34	130	−170	−436	26	−56	−888	−460	−116	−554	−1372	−378	−1658	−418	−852	286	−1528	−760	678	486	−3262

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 13, 17, 221]: [1, 5, 6, 66] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 48 = 1 x 48 (phi(ord eps) x dim S_k^new); 7 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(221, 4)[1]   # 221.4.a.b

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 221:4 1000